Hands of the clock, Revisited.

Lemma. $$ \left| e^{i\,\alpha} - e^{i\,\beta} \right|^2 = \\ \left| \; \left[ \cos(\alpha) + i\,\sin(\alpha)\right] - \left[\cos(\beta) + i\,\sin(\beta) \right] \; \right|^2 = \\ \left[ \cos(\alpha) - \cos(\beta) \right]^2 + \left[ \sin(\alpha) - \sin(\beta) \right]^2 = \\ \cos^2(\alpha) - 2\cos(\alpha)\cos(\beta) + \cos^2(\beta) + \sin^2(\alpha) - 2\sin(\alpha)\sin(\beta) + \sin^2(\beta) = \\ 2 - 2\cos(\alpha-\beta) $$ Least Squares Method. Two possibilities: $$ \left| e^{i(\theta+2\pi/3)} - e^{i12\theta} \right|^2 + \left| e^{i(\theta-2\pi/3)} - e^{i720\theta} \right|^2 = \mbox{minimum}(\theta) \\ \left| e^{i(\theta-2\pi/3)} - e^{i12\theta} \right|^2 + \left| e^{i(\theta+2\pi/3)} - e^{i720\theta} \right|^2 = \mbox{minimum}(\theta) $$ Whatever minimum is the smallest. With the lemma: $$ 2 - 2\cos(\theta+2\pi/3-12\theta) + 2 - 2\cos(\theta-2\pi/3-720\theta) = \mbox{minimum}(\theta) \\ 2 - 2\cos(\theta-2\pi/3-12\theta) + 2 - 2\cos(\theta+2\pi/3-720\theta) = \mbox{minimum}(\theta) $$ Brute force algorithm with sampling $\Delta\theta = 2\pi/(60\times 720)$ :

program klok;

function minimum(theta : double; teken : integer) : double;
begin
  minimum := 2 - 2*cos(theta+teken*2*pi/3-12*theta)
           + 2 - 2*cos(theta-teken*2*pi/3-720*theta);
end;

function normal(theta : double) : double;
var
  OK : boolean;
  v : double;
begin
  v := theta;
  OK := false;
  while not OK do
  begin
    OK := (0 <= v) and (v < 2*pi);
    v := v - 2*pi;
  end;
  normal := v + 2*pi;
end;

procedure brute_force(teken : integer);
var
  k : integer;
  M,min,p,w : double;
begin
  min := 8; w := 0;
  for k := 0 to 43200-1 do
  begin
    p := k*2*pi/43200;
    M := minimum(p,teken);
    if min > M then
    begin
      w := p;
      min := M;
    end;
  end;
  Writeln('Minimum =',min);
  Writeln('H =',w,' <',w+2*pi/43200);
  Writeln('M =',normal(12*w),' =',w+teken*2*pi/3);
  Writeln('S =',normal(720*w),' =',w-teken*2*pi/3);
  Writeln;
end;

begin
  brute_force(+1);
  brute_force(-1);
end.

Output (in radians):

Minimum = 1.32888417836478E-0004
H = 3.04690854166910E+0000 < 3.04705398577343E+0000
M = 5.14697596413128E+0000 = 5.14130364406230E+0000
S = 9.42477796078314E-0001 = 9.52513439275905E-0001

Minimum = 1.32888417839328E-0004
H = 3.23627676551049E+0000 < 3.23642220961482E+0000
M = 1.13620934304831E+0000 = 1.14188166311729E+0000
S = 5.34070751109978E+0000 = 5.33067186790368E+0000

Differentiation of the minimum functions, resulting in two equations: $$ 11\sin(11\theta-2\pi/3) + 719\sin(719\theta+2\pi/3) = 0 \\ 11\sin(11\theta+2\pi/3) + 719\sin(719\theta-2\pi/3) = 0 $$ Solving these with MAPLE, resulting in numerical refinement of the above:

Digits := 50;
fsolve(11*sin(11*theta-2*Pi/3)+719*sin(719*theta+2*Pi/3)=0,theta=3.0469..3.0470);
fsolve(11*sin(11*theta+2*Pi/3)+719*sin(719*theta-2*Pi/3)=0,theta=3.2362..3.2364);

Giving (Hours hand in radians):

3.0469223755142191756046177765785671073381009266460
3.2362629316653673013206689899804386610562378721043

Don't know which one of the two is the best. It looks like the minima are exactly the same, namely:

0.0000339325557414428099817512048046050292710528311